User blog:Scorcher007/S - Large Countable Ordinal Notation. Chapter I, Up to KPm.
This notation not well-ordered, but well-formed. SLCON is a partially defined way of briefly expression of large countable ordinals. Base terms example (comparison with the terms used in OCF): ω = Sσ(0) Ω = Sσ(1) Ω2 = Sσ(2) I = Sσ'1(0) I2 = Sσ'1(1) e.t.c. Some notes: n∈O - means ∀n∈O∃(n terms or f(n) terms), used in defenition of collection ordinals. n ω = ω1CK = 1st П2-reflecting 2nd admissible > ω = ω2CK = 2nd П2-reflecting limit of (n<ω)-th admissible = sup(ωnCK)|n<ω = 1st П1-reflecting onto П2-reflecting; limit of (ω+(n<ω))-th admissible = sup(ωω+nCK)|n<ω = 2nd П1-reflecting onto П2-reflecting 1st fixed point limits of admissible = α↦ωαCK = 1st П1-reflecting onto П1-reflecting onto П2-reflecting = 2-П1-reflecting onto П2-reflecting 1st fixed point of fixed point limits of admissible = 3-П1-reflecting onto П2-reflecting 1st hyper-fixed point limits of admissible = 1st hyper-П1-reflecting onto П2-reflecting 1st recursively inaccessible = 1st П2-reflecting that is П1-reflecting onto П2-reflecting 2nd recursively inaccessible = 2nd П2-reflecting that is П1-reflecting onto П2-reflecting limit of (n<ω)-th inaccessible = 1st П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting) limit of (ω+(n<ω))-th inaccessible = 2nd П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting) 1st fixed point limits of inaccessible = 1st 2-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting) 1st fixed point of fixed point limits of inaccessible = 1st 3-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting) 1st hyper-fixed point limits of inaccessible = 1st hyper-П1-reflecting onto (П2-reflecting that is П1-reflecting onto П2-reflecting) 1st recursively 2-inaccessible = 1st П2-reflecting that is П1-reflecting onto П1-reflecting onto П2-reflecting =1st П2-reflecting that is 2-П1-reflecting onto П2-reflecting 1st recursively hyper-inaccessible = 1st П2-reflecting that is hyper-П1-reflecting onto П2-reflecting 1st recursively Mahlo = 1st П2-reflecting onto П2-reflecting; Reference zoo - means refer to http://www.madore.org/~david/math/ordinal-zoo.pdf (Madore D., Zoo of ordinals, 2017) S - Large Countable Ordinal Notation (SLCON). Chapter I, Up to KPm. 0 SLCON - notation of x ordinal definition of x ordinal; PTO; (other function limit); {reference} DAN separator s(n,n{...}2) compration 1 ω = Sσ = Sσ(0) = Sσ'0 = Sσ'0(0) = S[SSσσ] = S[SS[SSSσσσ]] = S[SS[SSS[SSSσσσσ]]] = S(1)σ(1) = SGg[Gg] 1st admissible, 1st transfinite number, limit of (n<ω)-th natural number, 1st П1-reflecting; PA; ACA0; (exAN limit, Kirby-Paris hydra limit, Goodstein function limit); {zoo 1.4, PTO - zoo 1.10} {1,,2} 2 Sσ(1) 1st admissible after ω, ω1CK, 1st П2-reflecting; ACA+BI; (EAN limit, Bachmann's OCF limit); {zoo 2.1, PTO - zoo 1.20} {1,,3} 3 Sσ(2) 2nd admissible after ω, ω2CK, 2nd П2-reflecting {1,,4} 4 Sσ(3) 3d admissible after ω, ω3CK, 3d П2-reflecting {1,,5} 5 Sσ(4) 4-th admissible after ω, ω4CK, 4-th П2-reflecting {1,,6} 6 Sσ(n∈ω) ∀n∈ω∃n-th admissible, ωnCK|n<ω; admissible; KPωr+IND; П11-CA0; (mEAN limit, Buchholz hydra limit) {1,,1,2} 7 Sσ(n<ω) limit of (n<ω)-th admissible, sup(ωnCK)|n<ω, 1st П1-reflecting onto П2-reflecting; П11-CA+BI; (Buchholz ω-hydra limit, Buchholz's OCF limit, Feferman's OCF limit); {zoo 2.2, PTO - zoo 1.21} {1,,2,2} 8 Sσ(ω) ω-th admissible, ωωCK {1,,3,2} 9 Sσ(ω+1) (ω+1)-th admissible, ωω+1CK {1,,4,2} 10 Sσ(ω+(n∈ω)) ∀n∈ω∃(ω+n)-th admissible, ωω+nCK|n<ω {1,,1,3} 11 Sσ(ω+(n<ω)) limit of (ω+(n<ω))-th admissible, sup(ωω+nCK)|n<ω, 2nd П1-reflecting onto П2-reflecting {1,,2,3} 12 Sσ(ω×2) (ω×2)-th admissible, ωω×2CK {1,,3,3} 13 Sσ(ω×2+1) (ω×2+1)-th admissible, ωω×2+1CK {1,,4,3} 14 Sσ(ω×(n∈ω)) ∀n∈ω∃(ω×n)-th admissible, ωω×nCK|n<ω {1,,1,1,2} 15 Sσ(ω×(n<ω)) limit of (ω×(n<ω))-th admissible, sup(ωω×nCK)|n<ω {1,,2,1,2} 16 Sσ(ω2) (ω2)-th admissible, ωω2CK {1,,3,1,2} 17 Sσ(ω2+1) (ω2+1)-th admissible, ωω2+1CK {1,,4,1,2} 18 Sσ(ω(n∈ω)) ∀n∈ω∃ωn-th admissible, ωωnCK|n<ω {1,,1{2}2} 19 Sσ(ω(n<ω)) limit of ω(n<ω)-th admissible, sup(ωωnCK)|n<ω {1,,2{2}2} 20 Sσ(ωω) (ωω)-th admissible, ωωωCK {1,,3{2}2} 21 Sσ(ωω+1) (ωω+1)-th admissible, ωωω+1CK {1,,4{2}2} 22 Sσ((n∈ω)ω) ∀n∈ω∃nω-th admissible, ωnωCK|n<ω; Δ12-CA {1,,1{1{1,,2}2}2} 23 Sσ((n<ω)ω) limit of (n<ω)ω-th admissible, sup(ωnωCK)|n<ω {1,,2{1{1,,2}2}2} 24 Sσ(ε0) (ε0)-th admissible, ωε0CK {1,,3{1{1,,2}2}2} 25 Sσ(ε0+1) (ε0+1)-th admissible, ωε0+1CK {1,,4{1{1,,2}2}2} 26 Sσ(n∈Sσ(1)) ∀n∈ω1CK∃n-th admissible, ωnCK|n<ω1CK {1,,1{1,,2}2} 27 Sσ(n